Bounding slopes of $p$-adic modular forms
| dc.creator | Smithline, Lawren | |
| dc.date | 2007-05-24 | |
| dc.date.accessioned | 2026-07-07T08:03:07Z | |
| dc.date.available | 2026-07-07T08:03:07Z | |
| dc.description | Let $p$ be prime, $N$ be a positive integer prime to $p$, and $k$ be an integer. Let $P_k(t)$ be the characteristic series for Atkin's $U$ operator as an endomorphism of $p$-adic overconvergent modular forms of tame level $N$ and weight $k$. Motivated by conjectures of Gouvea and Mazur, we strengthen Wan's congruence between coefficients of $P_k$ and $P_{k'}$ for $k'$ close $p$-adically to $k$. For $p-1 | 12$, $N = 1$, $k = 0$, we compute a matrix for $U$ whose entries are coefficients in the power series of a rational function of two variables. We apply this computation to show for $p = 3$ a parabola below the Newton polygon $N_0$ of $P_0$, which coincides with $N_0$ infinitely often. As a consequence, we find a polygonal curve above $N_0$. This tightest bound on $N_0$ yields the strongest congruences between coefficients of $P_0$ and $P_k$ for $k$ of large 3-adic valuation. | |
| dc.description | 15 pages. June 2001 preprint | |
| dc.identifier | https://arxiv.org/abs/0705.3614 | |
| dc.identifier | http://arxiv.org/abs/0705.3614 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/129466 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G18 | |
| dc.title | Bounding slopes of $p$-adic modular forms | |
| dc.type | text |